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This paper gives probabilistic analyses of two kinds of multidimensional bin packing problems: vector packing and rectangle packing. In the vector packing problem each of the d dimensions can be interpreted as a resource. A given object i consumes aij units of the jth resource, and the objects packed in any given bin may not collectively consume more than one unit of any resource. Subject to this constraint, the objects are to be packed into a minimum number of bins. The rectangle packing problem is more geometric in character. The ith object is a d-dimensional box whose jth side is of length aij, and the goal is to pack the objects into a minimum number of cubical boxes of side 1. We study these problems on the assumption that the aij are drawn independently from the uniform distribution over [0,1]. We study a vector packing heuristic called VPACK that tries to place two objects in each bin and a rectangle packing heuristic called RPACK that tries to pack one object into each of the 2d corners of each bin. We show that each of these heuristics tends to produce packings in which very little of the capacity of the bins is wasted. In the case of rectangle packing, we show that the results can be extended to a wide class of distributions of the piece sizes.